Clubs, leadership, athletics, and academics are all important.  Remember the the average USAFA cadet had somewhere around a 3.8 GPA in high school.  Academics (GPA, class rank, SAT/ACT scores) count for 60% of the admissions process, so that is the most important.  However, you cannot neglect the other aspects either.

For congressional nominations, connections are NOT required.  Each congressman will usually put together a committee to review and interview each person who sends them a nomination application.  They then select the best candidates.  If you want an in depth description, you can go to serviceacademyforums.com or talk.collegeconfidential.com and find the "nominations" threads under the Service Academy forums.